Problem: Simplify and expand the following expression: $ \dfrac{3x + 6}{5x + 6}+\dfrac{x + 8}{x + 2} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5x + 6)(x + 2)$ Multiply the first term by $\dfrac{x + 2}{x + 2}$ $ \begin{align*} \dfrac{3x + 6}{5x + 6} \times \dfrac{x + 2}{x + 2} & = \dfrac{(3x + 6)(x + 2)}{(5x + 6)(x + 2)} \\ & = \dfrac{3x^2 + 12x + 12}{(5x + 6)(x + 2)}\end{align*} $ Multiply the second term by $\dfrac{5x + 6}{5x + 6}$ $ \begin{align*} \dfrac{x + 8}{x + 2} \times \dfrac{5x + 6}{5x + 6} & = \dfrac{(x + 8)(5x + 6)}{(x + 2)(5x + 6)} \\ & = \dfrac{5x^2 + 46x + 48}{(x + 2)(5x + 6)}\end{align*} $ Now we have: $ = \dfrac{3x^2 + 12x + 12}{(5x + 6)(x + 2)} + \dfrac{5x^2 + 46x + 48}{(x + 2)(5x + 6)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{3x^2 + 12x + 12 + 5x^2 + 46x + 48}{(5x + 6)(x + 2)} $ $ = \dfrac{8x^2 + 58x + 60}{(5x + 6)(x + 2)}$ Expand the denominator: $ = \dfrac{8x^2 + 58x + 60}{5x^2 + 16x + 12}$